Electron-electron coalescence and interelectronic log-moments in atomic systems

Abstract

A measure of the spatial electron-electron coalescence in atomic and molecular systems is given by the local quantity 〈δ(u)〉,u = r₁₂ being the interelectronic vector and δ the three-dimensional Dirac delta function. Here it is argued that there exists a functionH(s,t), s > t > — 3, which depends on two interelectronic moments 〈u^α〉 and plays an important role in the electron correlation problem. In particular it is rigorously fulfilled that for an arbitrary but fixeds,

$$\mathop {\lim }\limits_{t \to - 3} H(s, t) = (1/4\pi ) \mathop {\lim }\limits_{t \to - 3} \left[ {\left( {t + 3} \right)\left\langle {u^t } \right\rangle } \right] = \left\langle {\delta (u)} \right\rangle .$$

Additionally, specific values of the functionH(s, t) are shown to be related to the log-moments 〈u^t ln u〉; e.g., the interelectronic mean logarithmic radius 〈ln u〉 is expressed in terms of H(0,0) in a simple manner. From a numerical point of view, the values of the quantities 〈u^t〉 and 〈u^t ln u〉 as well as the values of the two-moment functionH(s,t) are calculated in the two-electron atoms with nuclear charge Z = 1,2,3,5 and 10 by means of the optimum 20-term Hylleraas-type wave functions. Finally, further bounds to 〈δ(u)〉 valid for the aforementioned two-electron atoms are found by means of log-moments.